Discontinuity-aware KAN-based physics-informed neural networks

This paper proposes Discontinuity-aware Physics-Informed Neural Networks (DPINNs), a novel framework that integrates adaptive Fourier-feature embeddings, a discontinuity-aware Kolmogorov-based architecture, mesh transformation, and learnable artificial viscosity to overcome spectral bias and instability, thereby achieving superior accuracy in solving partial differential equations with sharp discontinuities and complex geometries.

The Gist

Imagine you are trying to teach a super-smart robot to predict how air flows around an airplane wing. This is a classic physics problem involving complex equations.

Usually, these equations are smooth, like a gentle rolling hill. Standard AI models (neural networks) are great at learning smooth hills. They are like artists who love painting soft, flowing watercolors.

The Problem: The "Cliff" Issue
However, in real life, air doesn't always flow smoothly. Sometimes, it hits a "wall" of pressure and creates a shockwave—a sudden, violent jump from smooth air to compressed air. In math terms, this is a discontinuity (a cliff).

If you ask a standard "watercolor artist" AI to paint a cliff, it gets confused. It tries to smooth out the edge to make it look nice, resulting in a blurry, wobbly mess. It can't handle the sharp jump. This is why current AI struggles to predict supersonic flight or explosions accurately.

The Solution: The "Discontinuity-Aware" AI (DPINN)
The authors of this paper built a new kind of AI called DPINN (Discontinuity-aware Physics-Informed Neural Network). Think of it as upgrading the artist from a watercolor painter to a graphic designer with a sharp chisel.

Here is how they built this "chisel" using four clever tricks:

1. The "Super-Resolution" Lens (Adaptive Fourier Embedding)

• The Analogy: Imagine trying to listen to a song. Standard AI hears the bass and the melody (low frequencies) but misses the tiny, high-pitched cymbal crashes (high frequencies) where the shockwaves live.
• The Fix: The authors gave the AI a special lens that can instantly tune into those high-pitched cymbal crashes. It allows the AI to "see" the sharp details of the shockwave instead of blurring them out.

2. The "Chisel" Artist (DKAN)

• The Analogy: Standard AI is built like a smooth, continuous curve. It physically cannot draw a sharp corner without breaking its own rules.
• The Fix: They swapped the standard AI brain for a Kolmogorov-Arnold Network (KAN). Imagine a standard AI is a single long rope; a KAN is a chain of small, flexible links. This new structure is mathematically designed to handle "jumps." It's like giving the artist a chisel that can carve a perfect 90-degree corner instead of trying to paint a curve.

3. The "Smart Map" (Mesh Transformation)

• The Analogy: Imagine trying to draw a detailed map of a tiny, crowded city street using a map of the whole country. You can't see the details because the map is too zoomed out.
• The Fix: The AI uses a "smart map" that stretches the area where the action is happening (the shockwave) and squishes the empty areas. This forces the AI to focus all its brainpower on the tricky parts of the problem, making the learning process much faster.

4. The "Smart Glue" (Learnable Artificial Viscosity)

• The Analogy: When a car hits a wall, it crumples. In math, this "crumpling" causes the AI to panic and oscillate (shake wildly). Engineers usually add "glue" (viscosity) to smooth out the crash so the math works, but too much glue makes the crash look fake and soft.
• The Fix: Instead of using a fixed amount of glue, this AI has a learnable glue. It acts like a smart mechanic: it only puts glue exactly where the crash is happening, and it learns how much glue is needed. If the crash is small, it uses a drop; if it's huge, it uses more. It learns this on the fly, keeping the solution stable without ruining the sharpness of the shockwave.

The Results: Why It Matters

The researchers tested this new AI on three tough challenges:

1. The Burgers' Equation: A simple math test for shockwaves.
2. Riemann Problems: Simulating explosions and shock tubes.
3. Airplane Wings: Simulating air flowing at supersonic speeds (faster than sound).

The Verdict:
While old AI models produced blurry, inaccurate results or failed completely, the new DPINN drew sharp, crisp shockwaves. It did this with fewer parameters (less memory needed) and higher accuracy than previous methods.

In Summary:
This paper teaches us how to stop AI from "smoothing over" the messy, violent, and sudden parts of physics. By giving the AI a better lens, a sharper brain, a focused map, and a smart glue gun, we can finally use AI to solve some of the most difficult problems in aerodynamics and engineering.

Technical Summary

1. Problem Statement

Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs) for forward and inverse problems. However, they struggle significantly with PDEs involving discontinuous solutions (e.g., shock waves in aerodynamics, explosions, multiphase flows).

• Spectral Bias: Standard neural networks (MLPs) assume smoothness, making them prone to spectral bias. They fail to capture high-frequency components and steep gradients, leading to the Gibbs phenomenon (non-physical oscillations) near discontinuities.
• Optimization Paradox: The presence of discontinuities induces oscillatory loss landscapes, preventing effective convergence and causing numerical instability.
• Limitations of Existing Solutions:
  • Physical modifications (e.g., global artificial viscosity) smooth discontinuities but sacrifice accuracy.
  • Architecture modifications (e.g., Transformers, Fourier features) often require massive parameter counts and computational resources, limiting them mostly to 1D cases.
  • Mesh-based methods (e.g., WENO) are accurate but computationally expensive for outer-loop tasks like optimization and control.

2. Methodology: Discontinuity-aware PINN (DPINN)

The authors propose DPINN, a hybrid framework that integrates architecture, loss, and physics modifications to resolve discontinuities efficiently. The method consists of four core components:

A. Adaptive Fourier-Feature Embedding (KAF)

To mitigate spectral bias and capture multi-frequency features without manual tuning:

• Instead of fixed Random Fourier Features (RFF), the authors introduce a Kolmogorov-Arnold Fourier (KAF) layer.
• It uses a learnable hybrid spectral correction mechanism combining low-frequency (GELU) and high-frequency (Fourier) components.
• High-frequency weights are initialized small and grow during training, allowing the network to adaptively learn steep gradients and discontinuities.

B. Discontinuity-aware Kolmogorov–Arnold Network (DKAN)

To model shock-wave properties with fewer parameters:

• Based on the Kolmogorov-Arnold representation theorem, the network replaces standard MLP layers with DKAN layers.
• The activation functions are a combination of Dynamic Tanh (DyT) and parameterized B-splines.
• This architecture allows the network to approximate both continuous functions and discontinuities (shocks) automatically, significantly reducing the number of trainable parameters compared to MLPs.

C. Mesh Transformation

To handle complex geometries (e.g., airfoils):

• A nonlinear, invertible mesh transformation maps the physical domain (body-fitted coordinates) to a uniform computational grid.
• This stretches high-resolution regions and compresses low-resolution ones, improving convergence speed and accuracy for problems with sharp spatial transitions.

D. Learnable Local Artificial Viscosity (AV)

To stabilize the solution near discontinuities without global smearing:

• An artificial viscosity term ($\mu \nabla^2 Q$) is added to the governing equations (Euler equations).
• Localization: Viscosity is applied only in regions identified by a shock sensor ($s(x)$) and a stagnation sensor (to distinguish shocks from stagnation points).
• Learnability: The magnitude of viscosity is controlled by a single learnable parameter ($w_\nu$) scaled by the spectral radius of the flux Jacobians. This allows the network to adapt the dissipation level during training, minimizing unnecessary diffusion in smooth regions.

3. Key Contributions

1. Novel Architecture: The integration of DKAN with adaptive Fourier features creates a parameter-efficient network capable of representing discontinuous functions, generalizing the Kolmogorov theorem to the discontinuous regime.
2. Adaptive Physics Modification: The development of learnable local artificial viscosity that dynamically adjusts based on the solution's complexity, balancing stability and accuracy better than fixed global or local viscosity methods.
3. Comprehensive Validation: The method is rigorously tested across 1D (Burgers' equation, Riemann problems) and 2D (transonic and supersonic airfoil flows) scenarios, demonstrating robustness in capturing normal shocks, bow shocks, and oblique shocks.
4. Efficiency: DPINN achieves superior accuracy with significantly fewer parameters (often <1/5th of MLP-based methods) while maintaining competitive inference times.

4. Results

The paper benchmarks DPINN against standard MLPs, KANs, and variants with global/local artificial viscosity across several test cases:

• 1D Inviscid Burgers' Equation:
  • DPINN achieved relative errors ($R_1, R_2$) below 1% (0.80% and 0.88%), whereas standard MLPs and KANs failed to converge or produced errors >26%.
  • It required ~50x less training time than MLPs to reach similar accuracy levels and used <1/5th the parameters.
• 1D Riemann Problems (Sod and Lax Shock Tubes):
  • DPINN accurately captured shock waves, expansion fans, and contact discontinuities.
  • Standard MLPs with local AV failed to resolve the stronger shocks in the Lax case, producing overly dissipative fields. DPINN maintained sharp shock profiles.
• 2D Transonic Airfoil Flow (NACA0012, Ma=0.7):
  • DPINN successfully captured the normal shock on the airfoil surface with errors <4%.
  • Conventional methods either missed the shock entirely or produced smoothed, low-resolution profiles.
  • Parameter Efficiency: DPINN used ~5,785 parameters compared to ~67,208 for MLP-based models, while achieving 8.65x higher L2 accuracy.
• 2D Supersonic Airfoil Flow (NACA0012, Ma=1.3):
  • DPINN correctly resolved both the detached bow shock and trailing-edge oblique shocks.
  • It learned a lower viscosity coefficient ($5.41 \times 10^{-4}$) compared to fixed viscosity methods ($1.25 \times 10^{-3}$), indicating less artificial error introduction.

5. Significance

This work represents a significant step forward in applying AI to computational fluid dynamics (CFD) for high-speed flows.

• Bridging the Gap: It effectively bridges the gap between the high accuracy of traditional numerical methods (like WENO) and the speed/flexibility of neural networks.
• Scalability: By drastically reducing the parameter count via KANs and using local rather than global viscosity, the method becomes viable for complex, multi-dimensional problems that were previously too computationally expensive for PINNs.
• Future Potential: The framework provides a foundation for solving inverse problems, optimization, and control in systems with shocks, though the authors note that extending the shock-sensor robustness to fully 3D unsteady flows remains a future challenge.

In summary, DPINN offers a robust, parameter-efficient, and highly accurate solution for PDEs with discontinuous solutions, overcoming the spectral bias and instability issues that have historically limited the application of PINNs in shock-dominated flows.